Model predictive control for PHEBs based on driving style recognition in car-following scenarios

Abstract

With the rapid growth of the new energy vehicle sector, plug-in hybrid electric buses (PHEBs) have become an important part of public transportation. This research proposes a model predictive control (MPC) strategy for PHEB in the car-following scenarios based on driving style detection, which comprehensively examines fuel economy and safety. The approach begins by defining and computing a set of driving style indicators, which are used to classify driver behavior and dynamically adjust the equivalent factor (EF) in the equivalent consumption minimization strategy (ECMS). Then, a total cost function is formulated, which includes the cost of safety and fuel consumption. This method considers the two costs together to obtain the optimal cost function. Furthermore, the proposed energy management strategy (EMS) is tested under two standard operating conditions UDDS and CLTC, and compared with the conventional ECMS. Simulation results illustrate that with sufficient safe following distances, this method is effective in reducing overall energy use by 8.96% for aggressive drivers and slightly less for conservative drivers. Hardware-in-the-loop (HIL) test results confirm that the achieved fuel economy is commensurate with theoretical expectations.

Keywords

car-following scenario model predictive control driving style recognition equivalent factor

Introduction

In recent years, mounting concerns over environmental degradation and dwindling fossil fuel reserves have spurred substantial investment in alternative transportation technologies, including electric vehicles, extended-range electric vehicles, and hybrid electric vehicles (HEVs).^1–4 Plug-in hybrid electric buses combine the advantages of traditional internal combustion engine vehicles and electric vehicles, offering enhanced fuel efficiency and greater operational range.⁵ For PHEBs, an ideal operating condition is one in which the stored electrical energy in the onboard battery is fully depleted by the end of a scheduled route.^6–8 To enhance the adoption and effectiveness of HEVs, the development of an efficient energy management strategy aimed at minimizing fuel use for PHEBs is essential.⁹

The allocation of torque among the various power sources in hybrid electric vehicles, including the battery pack, electric motor (EM), and engine, presents a significant challenge for both research and industry.¹⁰ This problem has motivated extensive research into EMS design,¹¹ with approaches broadly falling into three categories: rule-based, optimization-based, and learning-based methods. Rule-based strategies depend heavily on the designers’ engineering expertise and experience.^12,13 Optimization methods can be further divided into global and real-time variants.^14–16 Global optimization techniques, such as dynamic programming (DP) and the Pontryagin Minimum Principle (PMP), yield theoretically optimal solutions but require full knowledge of future driving cycles.^17,18 In contrast, real-time optimization frameworks focus on control signals for a specific current or hourly period. Key methods in this category include MPC and ECMS, which formulate the optimal control problem as a constrained optimization problem.^19–22 Within ECMS, the equivalent factor serves as a critical tuning parameter that governs the trade-off between electrical and fuel energy usage, directly influencing system efficiency.²³ To enhance adaptability, recent studies have proposed dynamic adjustment mechanisms for the equivalent factor based on information such as driving conditions. For instance, an adaptive ECMS (A-ECMS) was developed which computes the equivalent factor in real time using powertrain energy variation and future traffic information, thereby improving fuel economy while maintaining robustness under varying operating conditions.²⁴ MPC, on the other hand, leverages predictions of driver intent and road conditions to dynamically optimize power split, thereby reducing overall energy consumption.^25,26 The performance of these methods is influenced by factors such as algorithm complexity, model accuracy, computational cost, and robustness. With the rise of artificial intelligence, learning-based approaches are increasingly gaining attention from researchers, integrating deep neural networks with traditional stochastic learning techniques to address energy management issues.²⁷

The majority of the methods mentioned above primarily focus on internal vehicle dynamics, often overlooking external contextual variables such as driving and traffic conditions. Among these factors, the driver’s driving style plays a crucial role and should not be neglected. A driver’s habits can significantly influence power distribution and impact energy efficiency.^28,29 Therefore, incorporating driving style into the design of energy management systems is essential for achieving optimal power distribution. Human drivers typically exhibit distinct behavioral tendencies that can be broadly classified into three archetypes: aggressive, conservative, and standard, each reflecting individual personality traits and situational responses.^30–32 To quantify these styles, multiple behavioral indicators are considered, such as throttle position, brake pedal input, vehicle speed, acceleration, deceleration, and fuel consumption. For instance, in Zhang and Xiong,³³ the rate of accelerator pedal variation is employed as an indicator across different driving cycles. Samani et al. examined and compared driver behaviors during manual driving and take-over scenarios. A comprehensive understanding of post-takeover behavior can be achieved by analyzing driving style, which emerges from the integration of various driving actions and contextual factors.^34,35 This study proposes an adaptive optimization strategy that incorporates driving style recognition via a fuzzy logic inference system.

It is important to recognize that a driver’s driving style is shaped not only by individual habits but also by dynamic interactions with surrounding traffic, particularly in car-following situations. Several studies have been conducted to enhance the understanding of EMS and vehicle-following control in HEVs.³⁶ Ruan et al.³⁷ designed a controller tailored for a variety of HEV congestion conditions, focusing on adjusting power distribution in real time to simultaneously enhance fuel efficiency, ride comfort, and tracking accuracy during following conditions. Xue et al.³⁸ proposed a transient optimal control strategy aimed at developing an adaptive ECMS that integrates performance in car-following contexts. This strategy seeks to optimize fuel efficiency, vehicle-following safety, and adaptability across various driving conditions. Xue and Jiao³⁹ implemented a cascaded speed control architecture based on parallel ETCS for HEVs operating in platooning-like scenarios in car-following situations. Nevertheless, the interdependence between driving safety in following contexts and the modulating effect of individual driving styles on energy consumption has received limited attention.

For PHEBs, EMS performance is inherently coupled with various external factors, including future traffic conditions and driver characteristics. To ensure energy-efficient control in car-following scenarios, the EMS must integrate adaptive cruise control with driving safety considerations, significantly increasing system complexity. Vehicle-to-vehicle (V2V) communication offers a viable pathway to address this: by providing previews of lead vehicle speed and inter-vehicle spacing, V2V data can empower MPC to proactively regulate powertrain output, improving fuel economy without compromising safe headway.⁴⁰ This paper propose a novel cost function that explicitly integrates a driving-style-dependent evaluation metric. Within an MPC framework, this formulation jointly optimizes car-following safety and fuel efficiency, overcoming a key limitation of prior strategies that assumed quasi-steady driving and ignored the frequent accelerations and decelerations typical of city traffic. However, MPC relies heavily on accurate prediction information, involves high computational burden, and sensitivity to parameter uncertainty pose challenges for embedded implementation. Therefore, despite its excellent performance in energy consumption optimization, it is still necessary to introduce ECMS as a complementary or online implementation solution to adapt to complex and varying real-world driving conditions for its computational efficiency and strong real-time capability. The primary objective of this study is to develop a driving-style-adaptive energy management strategy that simultaneously ensures safe vehicle following and optimal fuel economy for PHEBs in urban traffic, and the following outlines the main contributions of this work:

(1) A driver style coefficient factor is designed to differentiate between three different driving styles, based on which the adjustment strategy of equivalent factor in ECMS is optimized.

(2) The total cost function including the safety cost function and the fuel economy cost function was constructed, and the optimal EMS was designed under the premise of ensuring driving safety.

(3) The MPC-based framework incorporating this cost structure is implemented and validated through both simulation and hardware-in-the-loop experiments.

The remainder of this paper is structured as follows. Section II introduces the model of the PHEB powertrain. In Section III, the construction of the cost functions and MPC architecture is detailed. The simulation results and experimental data are analyzed in Section IV. Section V concludes the study and outlines future directions.

PHEB model descriptions

Unlike conventional vehicles powered by internal combustion engines, HEVs offer higher fuel efficiency by harnessing two complementary power sources: the electric motor and the engine. With a larger battery capacity, PHEBs can more efficiently utilize battery energy, thereby leading to notable reductions in both fuel usage and operational expenses. This study centers on a PHEB equipped with a coaxial parallel configuration, which enables the engine’s power and the electric motor’s output to be transmitted through the same drive shaft. Additionally, the electric machine can function as either a motor or a generator, depending on the operating mode.⁴¹ The bus’s structural design is depicted in Figure 1, with a detailed presentation of the essential subsystems provided in this section.

Figure 1.

The schematic overview of the PHEB.

Vehicle longitudinal dynamics

Regarding the amount of fuel that automobiles use, longitudinal dynamics are the most important influencing factor. While lateral dynamics and steering maneuvers do contribute to energy usage, their effects are comparatively minor, and modeling them accurately involves considerable complexity. Therefore, the formulation which expresses the relationship between the driving torque and the coupling torque produced by the engine and the motor will be formulated as:

T_{w} = (T_{e} + T_{m}) i_{g} i_{0} η_{T} + T_{b}

where T_w denotes the wheel driving torque, T_e and T_m are the torque of the engine and motor, respectively. T_b represents the braking torque, η_T is the efficiency of the transmission system, i_g and i₀ correspond to the gear ratio of the gearbox and the final drive ratio, respectively. Table 1 lists the vehicle’s primary characteristics. Also, T_w can be expressed as:

T_{w} = [mg f_{r} \cos θ + \frac{1}{2} C_{d} ρ_{air} A_{f} V_{veh}^{2} + mg \sin θ + δ m \frac{d V_{veh}}{dt}] \cdot r

where g is the gravitational acceleration, θ is the road slope, ρ_air is the air density, δ is the rotational mass coefficient, and V_veh denotes the vehicle’s velocity.

Table 1.

PHEB’s primary characteristics.

Component	Parameter	Value
Engine (YC6L330–50) machine	Type	6.5 L, diesel engine
	Maximum torque	1150 Nm @ 2200 rpm
	Maximum power	250 kW
	Type	PMSM
	Maximum torque	2800 Nm
AMT	Gear ratios	6.71/3.77/2.26/1.44/1.0/0.77
Battery	Capacity	26.6 kWh
	Voltage	390 V
f _r	Rolling resistance	0.019
C _d	Air drag coefficient	0.6
A _f	Front area	6.05 m²
m	Mass	16,500 kg
r	Tire radius	0.512 m

Engine model

Improving fuel efficiency remains a key priority for most PHEBs, as even modest gains can lead to substantial reductions in operational expenses and enhance market competitiveness. Ignoring the complex dynamic characteristics, the transient fuel consumption $\overset{\cdot}{m_{f}}$ will be reckoned as:

\overset{\cdot}{m_{f}} = \frac{T_{e} a_{e} b_{e}}{367.1 ρ_{f} g}

where ω_e represents the engine’s rotational speed, ρ_f accounts for the fuel density, and b_e is the fuel consumption rate. As seen by the fuel usage contour map in Figure 2, b_e at different torque and speeds can be found.

Figure 2.

Fuel usage contour map of the equipped diesel engine.

Motor model

In this study, a permanent magnet synchronous motor is selected, this kind of motor can be used as both a traction motor and a generator.^42,43 The power consumption of a motor used as a generator can be estimated as follows:

P_{m} = ω_{m} T_{m} / η_{g}

The power consumption of the motor, while it operates as a traction motor, is best stated as:

P_{m} = ω_{m} T_{m} η_{m}

where P_m is the motor power, while ω_m, η_m, and η_g are the motor angular velocity, the motor efficiency, and the generator efficiency, respectively. η_m is the efficiency factor from the motor efficiency map.

Battery model

The battery is an essential part of the overall hybrid energy management system, not only supporting the internal combustion engine but also enabling all-electric propulsion and facilitating regenerative energy recovery. Various approaches have been proposed for modeling to enhance battery efficiency. Meanwhile, the state of power, the chemical reaction and thermal effect are usually neglected to simplify the whole battery model in this paper.⁴⁴ The equivalent circuit equation is given as:

V_{L} = V_{oc} - R_{0} I_{0}

where V_L, V_oc, I₀, and R₀ are the voltage of battery terminals, open-circuit voltage, battery current, and internal resistance, respectively.

The battery SOC dynamic can be expressed as:

S \overset{\cdot}{O} C = \frac{V_{oc} - \sqrt{V_{oc}^{2} - 4 R_{0} P_{batt}}}{2 R_{0} Q_{batt}}

where Q_batt is the battery charge capacity, and P_batt denotes the power output.

Figure 3 illustrates the relationships among battery SOC, internal resistance, and open-circuit voltage (OCV), with the effects of operating temperature and battery aging neglected for simplicity. Specifically, the blue dashed curve represents the internal resistance during discharging as a function of SOC, the red dashed curve shows the internal resistance during charging versus SOC, and the green solid curve depicts the open-circuit voltage.

Figure 3.

Variation of battery internal resistance (during charging and discharging) and open-circuit voltage with SOC.

Energy management strategy development

Efficiently coordinating power delivery between the internal combustion engine and the electric motor remains a central challenge in hybrid energy management, as it directly governs the vehicle’s overall fuel economy.⁴⁵ Although urban buses typically operate on fixed routes, their energy consumption is still significantly affected by driver behavior and local road characteristics. As a result, accounting for driving style is an essential component of any effective EMS. In dense city traffic, buses frequently operate in car-following mode, often spending extended periods tracking a lead vehicle. Recognizing this operational reality, the present work introduces a unified control framework that jointly optimizes safety and energy efficiency within a single car-following model.

The proposed framework and the experimental validation method are both illustrated in Figure 4. The whole framework begins with real-time driving style recognition based on vehicle kinematic data, which dynamically informs the adaptation of the ECMS via a driving-style-dependent equivalent factor. The adapted energy management objective is then embedded within a MPC architecture that simultaneously optimizes fuel economy and car-following safety through a composite cost function. Finally, the entire control strategy is validated through co-simulation under standardized drive cycles and HIL experiments.

Figure 4.

The overarching framework of the proposed approach.

Car-following model

A preceding bus usually drives ahead while a host bus follows behind, which constitutes the car following situation in this study, as shown in Figure 5. The host vehicle is equipped with V2V communication devices for data reception and a radar for determining the inter-vehicle separation from the bus in front. d_h, v_h, and a_h are the host vehicle’s position, velocity, and acceleration, respectively. d_p, v_p, and a_p are the position, velocity, and acceleration of the preceding bus, respectively. To ensure that increased velocity allows for adaptive changes in the inter-vehicular distance, as seen below, the appropriate inter-vehicle distance d_des used to be calculated by

d_{des} = T_{h} v_{h} + d_{0}

where d₀ stands for the distance at a standstill, and T_h is the headway time.

Figure 5.

Schematic diagram of the car-following model.

Thus, the relative velocity and distance error can be described as follows:

{\begin{matrix} d_{r} = d_{p} - d_{h} - d_{des} \\ v_{r} = v_{p} - v_{h} \end{matrix}

where d_r is the relative distance error, and v_r represents the relative velocity error.

Dynamic models of d_r and v_r are defined by

{\begin{matrix} {\overset{\cdot}{d}}_{r} & = v_{p} - v_{h} - T_{h} a_{h} \\ {\overset{\cdot}{v}}_{r} & = a_{p} - a_{h} \end{matrix}

The aim of car-following control is to uphold the vehicle’s fundamental tracking capability while giving priority to driving comfort, safety, and fuel efficiency. By adjusting the inter-vehicle distance to stay within the desired control range and synchronizing with the speed and acceleration of the lead vehicle, the likelihood of safety-critical events can be markedly lowered, thereby improving overall vehicle safety.

The two characteristic parameters, relative distance error d_r and relative speed error v_r can be used to evaluate the tracking performance of the rear vehicle at a safe following range. Also, the driving style can be judged by the acceleration a_h and the sudden change in acceleration. The amount of change in host bus acceleration $Δ a_{h}$ is used as the control variable to construct the car-following model, and the three state variables are relative distance error d_r, relative speed error v_r, and acceleration a_h. Furthermore, the preceding vehicle’s acceleration a_p is expected to remain constant during the projected horizontal interval.

Driving style recognition

Driving style refers to the manner in which a driver operates a vehicle under varying traffic and weather conditions. Given that the driving environment and road conditions remain relatively consistent under similar operational circumstances, the driver’s individual style becomes a key determinant in the car-following process. The sequential positions of the host vehicle along the roadway are depicted in Figure 6.

Figure 6.

The sampling data and sampling points for the host vehicle.

This study examines differences in car-following patterns and maps them onto key variables that characterize driving style. Using this set of indicators, the distinct driving styles of individual drivers can be effectively captured. In particular, within car-following contexts, acceleration and speed profiles provide more accurate and discernible reflections of a driver’s behavioral tendencies.

Based on the dependency between the test data and the driving styles, the mean, maximum and minimum values of relative speed, the standard deviation and derivative of acceleration were finally selected as inputs for driving style recognition, which can be calculated as follows:

v_{avg} = d_{h} / (t (N) - t (1))

where v_avg represents the average speed of the host vehicle. It is well known that speed, acceleration, throttle opening and brake pedal position serve as more effective modes to capture the relationship between driver driving style and fuel economy. By analyzing the speed change degree in the process of bus driving, the driver style recognition coefficient is proposed, and the accuracy and reliability of the driver style recognition coefficient are verified. The degree of impact I(t) and the standard deviation of which are defined as follows:

\begin{matrix} I (t) = \frac{d^{2} v_{h} (t)}{d t^{2}} = \frac{d a_{h} (t)}{dt} \\ I_{std} = \sqrt{\sum_{k = 1}^{N - 1} {((I (k) + I (k + 1)) / 2 - I_{avg})}^{2} / (N - 1)} \end{matrix}

where I_avg and I_std are the average and standard deviation of the degree of impact, respectively.

The driving style recognition factor R_driver is defined as follows:

R_{driver} = I_{std} / I_{avg}

The steps of driving style recognition are as follows:

(1) At each sampling instant t_k, vehicle speed data are collected over a predefined time window, based on which the average speed, average impact I_avg, average acceleration, etc. are derived.

(2) The standard deviation of the impact over this length of time is computed and combined with I_avg to yield the driving style recognition factor R_driver.

(3) The value of R_driver is then compared against two constants R_agg and R_norm. When R_driver is less than R_norm, it means that the driver is classified as exhibiting a conservative style. When R_driver is between R_norm and R_agg, it indicates that the behavior corresponds to a standard (or habitual) driving pattern. When R_driver is greater than R_agg, it represents that the current driving style is more aggressive.

(4) Here, the constant R_agg represents the aggressive driving style threshold and R_norm marks the standard driving style. Their respective values are 1 and 0.5.

In the proposed identification approach, three criteria are employed to classify the intensity of different driving styles. These criteria reflect not just how readily a driver accelerates or decelerates, but also how sensitively they respond to speed fluctuations and how their instantaneous acceleration deviates from the average. Through these factors, the recognition process can provide insight into individual driving patterns and reveal driving habits and preferences. This comprehensive evaluation contributes to the understanding of driver-vehicle interactions and provides valuable information for adaptive control strategies to improve energy efficiency and optimize performance. The three criteria are expressed as

β = {1, 3, 5}

where the level value of 5 represents the aggressive driving style, the level value of 3 denotes the standard one, and the level value of 1 is the conservative style.

In addition, a strategy for the transformation of the stylistic dimension is proposed as follows:

β (t + T) = β (\frac{1}{N} \sum_{i = 1}^{N} X (t + i T_{0}))

where X denotes the sampling vectors, N is the number of sampling vectors, T₀ and T are the sampling and update periods, respectively.

Driving safety indicator

The permissible distance range for the following vehicles should vary at maximum and minimum distances. Based on Li et al.,⁴⁰ the following bus’s permissible tracking range L should be limited within a reasonable interval:

L = d_{r} - d_{h}

The minimal and maximal distances, which are denoted by L_min and L_max, can be formulated as:

\begin{matrix} L_{min} (k) = 2 + 0.5 \cdot v_{h} (k) + 0.0625 \cdot {v_{h}}^{2} (k) \\ L_{max} (k) = 10 + v_{h} (k) + 0.0825 \cdot {v_{h}}^{2} (k) \end{matrix}

Meanwhile, an ideal following distance zone is determined, which can be described by

\begin{matrix} L_{ide}^{up} (k) = 0.4 \cdot L_{max} (k) + 0.6 \cdot L_{min} (k) \\ L_{ide}^{low} (k) = 0.4 \cdot L_{min} (k) + 0.6 \cdot L_{max} (k) \end{matrix}

where the ideal following distance zone’s lower and upper boundaries are indicated by superscripts low and up.

Driving safety costs can be estimated as follows:

J_{s} = {\begin{matrix} inf L (k) < L_{min} (k) \\ 0.2 \cdot \tan (\frac{π}{2} \cdot \frac{L (k) - L_{ide}^{low} (k)}{L_{min} (k) - L_{ide}^{low} (k)}) L_{min} \leq L (k) < L_{ide}^{low} (k) \\ L (k) - | \frac{L_{ide}^{low} (k) + L_{ide}^{up} (k)}{2} | L_{ide}^{low} (k) \leq L (k) \leq L_{ide}^{up} (k) \\ 2 \cdot {(L (k) - L_{ide}^{up} (k))}^{2} L_{ide}^{up} (k) < L (k) \leq L_{max} (k) \\ 2 \cdot {(L (k) - L_{ide}^{up} (k))}^{2} + 100 \cdot {(L (k) - L_{max} (k))}^{2} L (k) > L_{max} (k) \end{matrix}

where J_s is the inter-vehicle driving safety indicator at time step k.

Fuel consumption indicator

The fuel economy is the most vital index in the EM system. Commonly adopted approaches typically rely on the PMP or ECMS. Accordingly, total energy consumption over the driving cycle is taken as the main indicator for evaluating system performance. In this formulation, the engine torque T_e and the motor torque T_m are the two optimization control variables u in this issue, while the battery SOC is the state variable x:

{\begin{matrix} x (t) = soc (t) \\ u (t) = {[T_{e} (t), T_{m} (t)]}^{T} \end{matrix}

The ECMS approach thus links fuel consumption with electrical energy usage through an equivalence factor. The instantaneous equivalent fuel consumption is obtained by adding the actual fuel mass flow rate to the fuel-equivalent of electricity consumption. This relationship is expressed by the following equation:

{\overset{\cdot}{m}}_{f, eqv} (t) = {\overset{\cdot}{m}}_{f} (t) + {\overset{\cdot}{m}}_{ress} (t)

where ${\overset{\cdot}{m}}_{f, ress}$ is the instantaneous equivalent fuel consumption, and ${\overset{\cdot}{m}}_{ress}$ denotes the energy consumed by the motor.

Through an analogy with an engine that consumes actual fuel, the instantaneous fuel consumption is defined as:

{\overset{\cdot}{m}}_{ress} (t) = \frac{s (t)}{Q_{lhv}} P_{batt} (t)

where s(t) represents the equivalent coefficient of oil-electric conversion, and Q_lhv is the fuel-low heating value. The EF needs to be dynamically adjusted in each driving cycle with the aim of enhancing the auxiliary role of the motor during high power demand or reducing excessive power generation during conservative driving. Thus, the continuous charging and optimal fuel efficiency of the hybrid system can be guaranteed to be maximized. Within the vehicle, a driving style recognition module is installed, which continually assesses the driver’s actions at every time step. The specific formulas are expressed as follows:

\begin{matrix} φ (β (t + T)) = 1 + {(\frac{β (t + T) - β_{tar}}{η})}^{γ} \\ {\begin{matrix} β_{tar} = \frac{β_{max} + β_{min}}{2} \\ γ = 2 n + 1, n \in N \end{matrix} \end{matrix}

where φ denotes the penalty coefficient associated with driving style, η is a constant, β_tar represents the standard driving style, and N is the number of sampling vectors.

Based on the process above, the EF can be rewritten as follows:

s^{*} = s_{0} + φ \times (s - s_{0})

where s* is the ideal equivalent factor, and s₀ is the initial equivalent factor. In this study, fuel efficiency performance corresponds to equivalent fuel consumption that includes both gasoline and electricity. Thus, the cost function for a typical MPC should be defined as:

J_{eco} = \int_{t_{0}}^{t_{f}} [{\overset{\cdot}{m}}_{f} (x (t), u (t), t) + \frac{s (t)}{Q_{ih v}} \times P_{batt} (t)] dt

Considering the characteristics of MPC, after discretizing the state equation, the cost function is defined as:

J_{eco} (k) = {\overset{\cdot}{m}}_{f} (x (k), u (k)) + \frac{s^{*} (k)}{Q_{ihv}} \times P_{batt} (k)

MPC-based energy management strategy

To manage the intricate balance between safety and fuel efficiency across different driving styles, a new cost function is introduced within the MPC framework. This formulation features an adaptive weighting scheme that adjusts in real time according to the driver’s identified behavior. In practice, when aggressive driving is detected, greater emphasis is placed on maintaining a safe following distance; conversely, eco-friendly driving patterns shift the priority toward minimizing fuel consumption.

The MPC controller is implemented with a prediction horizon $N_{p} = 20$ steps and a control horizon of five steps, corresponding to a total prediction window of 2 s and a control update interval of 0.1 s, consistent with the vehicle’s sampling period $T_{0} = 0.1$ s. The prediction model assumes the preceding vehicle’s acceleration remains constant over the prediction horizon, which provides sufficient accuracy for short-term car-following scenarios in urban traffic while maintaining computational tractability. These settings were selected through preliminary tuning to balance performance, robustness. The baseline weighting coefficients are established through a rigorous multi-objective optimization procedure using offline co-simulation with the vehicle model. A Pareto front is generated from simulations spanning a range of representative driving scenarios, enabling the identification of non-dominated solutions that jointly consider safety margin, fuel use, and ride comfort. An initial set of optimal weights is then chosen from this front as a balanced compromise, ensuring consistently acceptable performance across all objectives.

The following is the formulation of the optimization problem of safety and fuel consumption at time step k at the edge of the prediction horizon:

\begin{matrix} J_{over} (k) = \sum_{t = k}^{k + N_{P} - 1} (J_{s} (k) + J_{eco} (k)) \times T_{0} \\ = \sum_{t = k}^{k + N_{p} - 1} {J_{s} (k) + {\overset{\cdot}{m}}_{f} (x (k), u (k)) + \frac{s^{*} (k)}{Q_{ihv}} \times P_{batt} (k)} \times T_{0} \end{matrix}

where N_p denotes the length of the receding horizon, which is usually 0.1 s. J_over is the overall cost function. MPC is taken to optimize the control problem as formulated below:

J_{over} (k) = min \sum_{t = k}^{k + N_{p} - 1} (J_{s} (k) + J_{eco} (k)) \cdot T_{0}

In the process of global optimization, due to the limitations of the whole vehicle system, some constraints should also be applied to limit the system states and control variables. These limitations can be summarized as follows:

{\begin{matrix} SO C_{min} \leq SOC (k) \leq SO C_{max} \\ P_{batt, min} \leq P_{batt} (k) \leq P_{batt, max} \\ T_{x, min} \leq T_{x} (k) \leq T_{x, max} \\ ω_{x, min} \leq ω_{x} (k) \leq ω_{x, max}, x = eng, mot, gen \\ a_{h, min} \leq a_{h, k} \leq a_{h, max} \\ L_{min} \leq L_{k} \leq L_{max} \end{matrix}

where the subscripts ‘min’ and ‘max’ stand for the minimum and maximum allowed values, respectively. The first equality is used to calculate the host bus’s speed at time step k + 1. Also, the SOC, battery power, and the speed or torque of engines, generators, and motors are all constrained by the following four inequalities, which are local restrictions. The last two inequalities are a global constraint, representing that the host vehicle needs to meet safety targets when following the preceding bus.

Simulation results and analysis

This section demonstrates how the predictive EMS, enhanced with driving style recognition, performs in car-following situations. The evaluation focuses on two key aspects: vehicle trackability and fuel efficiency. All simulations are carried out in MATLAB/Simulink, where a detailed PHEB model has been developed. To ensure reproducibility and experimental clarity, the preceding vehicle’s speed trajectories are derived directly from two widely adopted standard drive cycles—the Urban Dynamometer Driving Schedule (UDDS) and the China Light-Duty Vehicle Test Cycle (CLTC)—which serve as benchmark test conditions for the lead vehicle in all car-following scenarios. The host bus is initialized with the SOC of 60%, a standstill following distance of 15 m, and zero relative velocity with respect to the lead vehicle. With driving safety in mind, the discussion centers on the bus’s tracking accuracy and its fuel consumption behavior under these systematically constructed scenarios. The same vehicle dynamics model and control period used in simulation are deployed in the hardware-in-the-loop tests, where the UDDS trajectory is replayed via CAN messages to emulate realistic input conditions, ensuring consistency between simulation and experimental validation.

Tracking capability

The host bus follows the preceding vehicle according to the target car-following strategy outlined earlier. Figures 7 and 8 present the results for the two driving cycles, including the car-following distances and the speeds of both the host and preceding vehicles. It can be observed that the host bus closely tracks the speed of the leading vehicle while rapidly converging to the desired following distance. Under both test conditions, the host vehicle’s speed profile closely matches that of the preceding vehicle, with a deviation of less than 10%, indicating effective tracking performance and a low safety cost. Figure 9 displays the absolute relative distance error over time. Throughout the cycles, the error remains within 6 m, demonstrating the proposed strategy’s strong tracking capability. Moreover, the error magnitude is consistently larger under aggressive driving styles than under conservative ones, consistent with the higher dynamic intensity and greater variability typically associated with aggressive maneuvers.

Figure 7.

Following distance and velocities at UDDS driving cycle.

Figure 8.

Following distance and velocities at CLTC driving cycle.

Figure 9.

Following distance error at the two test conditions: (a) UDDS driving cycle and (b) CLTC driving cycle.

Comparison with the proposed strategy for various drivers

This study evaluates the proposed EMS primarily in terms of fuel efficiency and car-following tracking performance. It should be noted that drivers with different styles exhibit distinct behaviors when maintaining a safe following distance. In such scenarios, the energy management technique outlined above can be applied to effectively distribute torque between the engine and motor in a satisfactory manner. To preserve drivability of the PHEB, the combined output torque from both power sources must satisfy the total demand torque of the hybrid powertrain. The results are presented in Figures 10 and 11, which show the engine and motor torque profiles under the two driving cycles, reflecting the control performance of the proposed strategy. The standard driving style serves as the reference case in our adaptive framework; therefore, the presented comparisons concentrate on the two representative extremes. From these figures, it can be observed that the motor plays a dominant role in meeting traction demands, particularly at low speeds, which enables more flexible adjustment of the engine’s operating points and helps avoid inefficient engine operation in low-efficiency regions. Meanwhile, compared to conservative drivers, aggressive drivers frequently experience abrupt acceleration and deceleration, which makes the throttle to open and close frequently, resulting in significant variations in the engine and motor torque outputs.

Figure 10.

The distribution of engine and motor torque at UDDS driving cycle.

Figure 11.

The distribution of engine and motor torque at CLTC driving cycle.

Battery SOC serves as a key indicator of electrical energy usage at each time step. The battery’s SOC curves of the conservative and aggressive drivers under two different driving cycles are displayed in Figure 12. The result clearly demonstrates the effect of driving style. Due to different driving habits, drivers with different driving styles have different values of the EF. Notably, the final SOC levels diverge between the two styles. Frequent acceleration and deceleration cause the battery’s SOC to degrade more quickly than other drivers. If the SOC drops too rapidly, the motor may lack sufficient capacity to provide the necessary torque for regulating the engine’s operating region, potentially compromising fuel efficiency.

Figure 12.

The SOC curves at UDDS and CLTC driving cycle when using ECMS.

The suggested approach fully utilizes electrical and gasoline energy to meet the vehicle’s power requirements based on the driver’s driving style. In the case of aggressive drivers, as shown in Figure 13, the rate of decline in SOC is effectively reduced compared to ECMS in UDDS conditions due to the ability to update the EF according to driving style. Although SOC trajectories under CLTC are not plotted separately, the energy consumption data in Table 2 confirm that the proposed strategy similarly moderates SOC depletion and improves overall efficiency in the CLTC scenario. The sum of the costs for the fuel and electricity used throughout the entire cycle makes up the total cost. It can be seen that there is a significant reduction in the total cost of the proposed strategy. It is evident that the proposed strategy lowers the overall cost compared to ECMS. Specifically, the strategy’s total cost is nearly 8.96% lower than that of ECMS, despite a slight rise in electric consumption. Therefore, the proposed approach has potential in terms of fuel economy, especially in car-following situations.

Figure 13.

Comparison of the battery SOC trajectory at UDDS driving cycle.

Table 2.

Comparison of the fuel economy.

Method	Fuel consumption (L)	Electric consumption (kWh)	Total cost (CNY)	Reduction (%)
ECMS	2.369	5.704	21.427	—
Proposed strategy	2.127	5.892	19.764	7.76

Hardware-in-the-loop test

HIL experiments are conducted to further evaluate the performance of the proposed control architecture. The HIL test setup, shown in Figure 14, comprises both hardware and software components. On the hardware side, the platform integrates power management modules, a TDK Lambda GEN30-50 programmable power supply, an NI Real-Time Simulator, and two PXI-8512 CAN bus interface boards. The software suite includes a human–computer interaction (HCI) display interface, a vehicle dynamics model, a communication module, and a load simulation unit. The HCI display platform, built on the NI VeriStand environment, enables the development of custom virtual instruments, real-time entry of driver control commands, and continuous monitoring of key system variables. The offline vehicle system model is first compiled and then deployed onto the NI Real-Time Simulator. Once loaded, the simulator streams real-time data on states and parameters to the upper-level computer’s monitoring platform for visualization and analysis.

Figure 14.

The framework of HIL test.

In the HIL test, the UDDS test driving cycle is chosen as the sample. The comparison results of engine torque, motor torque, and speed are shown in Figure 15. As can be seen, the velocity curves are in good agreement with the simulated ones. Additionally, both the motor torque and the engine torque fit the simulation results well. There are only 4.7% and 6.1% errors in torque values due to the difference between simulation and HIL testing in signal transmission and processing. The HIL test results show that the proposed strategy is consistent with the original design intention.

Figure 15.

The HIL test results.

Conclusion

This work presents a multi-objective MPC framework that jointly accounts for safety and fuel consumption by incorporating real-time driving style identification. The strategy is designed to enhance the fuel efficiency of PHEBs during car-following maneuvers while ensuring consistent safety margins across diverse driver behaviors. To support reliable style classification, a driving style coefficient is introduced. Upon identification, this coefficient dynamically tunes the equivalence factor in the cost function, thereby adapting the fuel consumption term to reflect the driver’s behavioral characteristics. Leveraging this adaptive mechanism, an MPC-based control architecture is formulated to explicitly explore the trade-off between energy economy and driving safety. The approach is tested under two representative driving conditions. Simulation outcomes show that, relative to baseline strategies ignoring driving style, the proposed method lowers the total cost by about 8.96%. HIL experiments further corroborate these findings, with results aligning well with the intended design goals and validating the practical viability of the strategy. Future work will incorporate ride comfort as an additional optimization criterion. Moreover, the influence of other PHEB components on energy management strategies and fuel consumption warrants further investigation, such as the clutch and transmission.

Footnotes

Handling Editor: Mingyang Zhang

ORCID iDs

Yifei Yang

Xiaodong Sun

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the ‘Smart Fruit’ Action Plan (Guangxi Science and Technology Achievement Transformation Plan) under Project ZG2503980022, and the Qinglan Project of Jiangsu province.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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